Finite Group Databasebeta

← #24127

Subgroups — #24127

Order 256.

544 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#552C211yesmaximal, characteristic
2C2×#479C211yesmaximal, characteristic, complemented
3C2×C4×Q16C211yesmaximal, characteristic, complemented
4#2536C211yesmaximal, characteristic
5#2036C211yesmaximal, characteristic, complemented
6#2035C211yesmaximal, characteristic, complemented
7#2001C211yesmaximal, characteristic, complemented
8#2001C211yesmaximal, characteristic, complemented
9#1991C211yesmaximal, characteristic, complemented
10#1954C211yesmaximal, characteristic
11#1873C211yesmaximal, characteristic, complemented
12#1872C211yesmaximal, characteristic, complemented
13#1854C211yesmaximal, characteristic, complemented
14#1852C211yesmaximal, characteristic, complemented
15#1805C211yesmaximal, characteristic, complemented
16C22×Q16C2211yescharacteristic
17#552—24no—
18#552—24no—
19C2×#126—21no—
20C2×#125—21no—
21C2×#125C2211yescharacteristic
22C2×#123—21no—
23C2×C22⋊Q8C2211yescharacteristic
24C2×C22⋊Q8C2211yescharacteristic
25C2×C22⋊Q8C2211yescharacteristic
26C2×C22⋊Q8C2211yescharacteristic
27C2×C4×Q8C2211yescharacteristic
28C2×C4×Q8C2211yescharacteristic
29C2×C4×D4C2211yescharacteristic
30C2×#117—21no—
31C2×#117C2211yescharacteristic
32C22×C4⋊C4C2211yescharacteristic
33#479—24no—
34C4×Q16—24no—
35C2×#107C2211yescharacteristic
36C2×#107C2211yescharacteristic
37C2×#106C2211yescharacteristic
38C2×C4⋊C8C2211yescharacteristic
39C2×#103C2211yescharacteristic
40C2×#103C2211yescharacteristic
41C2×#103C2211yescharacteristic
42C2×#103C2211yescharacteristic
43C2×#103C2211yescharacteristic
44C2×#103C2211yescharacteristic
45C2×C22⋊C8C2211yescharacteristic
46C2×C22⋊C8C2211yescharacteristic
47C2×C22⋊C8C2211yescharacteristic
48C2×C22⋊C8C2211yescharacteristic
49C2×C4×C8C2211yescharacteristic
50#398—21no—
51#394—21no—
52#388—21no—
53#388—21no—
54#382C2211yescharacteristic
55#380C2211yescharacteristic
56#380C2211yescharacteristic
57#340C2211yescharacteristic
58#340C2211yescharacteristic
59#340C2211yescharacteristic
60#340C2211yescharacteristic
61#340C2211yescharacteristic
62#340C2211yescharacteristic
63C22×Q8C2311yescharacteristic
64C22×Q8C2311yescharacteristic
65C22×D4D411yescharacteristic
66C23×C4—21no—
67C23×C4C2311yescharacteristic
68C23×C4C2311yescharacteristic
69C2×Q16—22no—
70C2×Q16—22no—
71C2×Q16—22no—
72C22×C8C2311yescharacteristic
73C22×C8C2311yescharacteristic
74C22×C8C2311yescharacteristic
75C22×C8C2311yescharacteristic
76#126—28no—
77#125—28no—
78#125—22no—
79#125D414yes—
80#123—28no—
81C22⋊Q8—22no—
82C22⋊Q8—22no—
83C22⋊Q8—22no—
84C22⋊Q8—22no—
85C22⋊Q8—22no—
86C22⋊Q8—22no—
87C22⋊Q8—41no—
88C22⋊Q8—41no—
89C4×Q8—22no—
90C4×Q8—22no—
91C4×Q8—22no—
92C4×Q8D414yes—
93C4×D4—22no—
94C4×D4—22no—
95#117—28no—
96#117—22no—
97#117—22no—
98C2×C4⋊C4—21no—
99C2×C4⋊C4—21no—
100C2×C4⋊C4—21no—
101C2×C4⋊C4—21no—
102C2×C4⋊C4—21no—
103C2×C4⋊C4—21no—
104C2×C4⋊C4—22no—
105C2×C4⋊C4—22no—
106C2×C4⋊C4—21no—
107C2×C4⋊C4—21no—
108C2×C4⋊C4—21no—
109C2×C4⋊C4D411yescharacteristic
110C2×C4⋊C4C2311yescharacteristic
111C2×C4⋊C4C2311yescharacteristic
112C2×C4⋊C4C2311yescharacteristic
113C2×C4⋊C4C2311yescharacteristic
114C2×C4⋊C4C2311yescharacteristic
115C2×C4⋊C4C2311yescharacteristic
116C2×C22⋊C4—21no—
117C2×C22⋊C4—21no—
118C2×C22⋊C4—21no—
119C2×C22⋊C4—21no—
120C2×C22⋊C4—21no—
121C2×C22⋊C4—21no—
122C2×C22⋊C4—21no—
123C2×C22⋊C4D411yescharacteristic
124C2×C22⋊C4D411yescharacteristic
125C2×C42—21no—
126C2×C42—21no—
127C2×C42—21no—
128C2×C42C2311yescharacteristic
129#107—22no—
130#107—22no—
131#106—22no—
132C4⋊C8—22no—
133#103—22no—
134#103—22no—
135#103—22no—
136#103—22no—
137#103—41no—
138#103—41no—
139C22⋊C8—22no—
140C22⋊C8—22no—
141C22⋊C8—41no—
142C22⋊C8—41no—
143C4×C8—22no—
144#95—21no—
145#95—21no—
146#95—21no—
147#95—21no—
148C24C2×D411yescharacteristic
149C24C2×D411yescharacteristic
150C2×Q8—21no—
151C2×Q8—21no—
152C2×Q8—21no—
153C2×Q8—21no—
154C2×Q8—21no—
155C2×Q8—21no—
156C2×Q8—21no—
157C2×Q8—21no—
158C2×Q8—21no—
159C2×Q8—21no—
160C2×Q8C2×D412yes—
161C2×Q8#4112yes—
162C2×D4—22no—
163C2×D4—22no—
164C2×D4—41no—
165C22×C4—24no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—21no—
171C22×C4—21no—
172C22×C4—21no—
173C22×C4—21no—
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—21no—
177C22×C4—21no—
178C22×C4—21no—
179C22×C4—21no—
180C22×C4—21no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—21no—
186C22×C4—42no—
187C22×C4—42no—
188C22×C4C2×D411yescharacteristic
189C22×C4C2×D411yescharacteristic
190C22×C4C2×D411yescharacteristic
191C22×C4C2×D411yescharacteristic
192C22×C4C2411yescharacteristic
193Q16—28no—
194C2×C8—21no—
195C2×C8—21no—
196C2×C8—21no—
197C2×C8—21no—
198C2×C8—21no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—21no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8#4112yes—
206C4⋊C4—24no—
207C4⋊C4—24no—
208C4⋊C4—24no—
209C4⋊C4—24no—
210C4⋊C4—24no—
211C4⋊C4—24no—
212C4⋊C4—22no—
213C4⋊C4—21no—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—21no—
217C4⋊C4—21no—
218C4⋊C4—21no—
219C4⋊C4—21no—
220C4⋊C4—21no—
221C4⋊C4—21no—
222C4⋊C4—42no—
223C4⋊C4—42no—
224C4⋊C4—42no—
225C4⋊C4C2×D412yes—
226C4⋊C4C2×D412yes—
227C4⋊C4C2×D412yes—
228C22⋊C4—24no—
229C22⋊C4—24no—
230C22⋊C4—24no—
231C22⋊C4—22no—
232C22⋊C4—22no—
233C22⋊C4—42no—
234C22⋊C4—42no—
235C22⋊C4—42no—
236C22⋊C4—42no—
237C42—24no—
238C42—24no—
239C42—42no—
240C42C2×D412yes—
241C42C2×D412yes—
242C23—21no—
243C23—21no—
244C23—21no—
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23#13512yes—
256C23C22×D411yescharacteristic
257Q8—22no—
258Q8—22no—
259Q8—22no—
260Q8—22no—
261Q8—41no—
262Q8—41no—
263Q8—41no—
264Q8—41no—
265D4—44no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—21no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—48no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—41no—
285C2×C4—41no—
286C2×C4—41no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—41no—
300C2×C4—41no—
301C2×C4—41no—
302C2×C4—41no—
303C2×C4—41no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—41no—
309C2×C4#12012yes—
310C2×C4#12012yes—
311C2×C4C22×D411yescharacteristic
312C2×C4C22×D411yescharacteristic
313C2×C4C2×#4111yescharacteristic
314C2×C4C2×#4111yescharacteristic
315C2×C4H(5,𝔽2)11yescharacteristic
316C2×C42−1+411yescharacteristic
317C8—22no—
318C8—41no—
319C8—41no—
320C8—41no—
321C22—22no—
322C22—22no—
323C22—22no—
324C22—41no—
325C22—41no—
326C22—41no—
327C22—41no—
328C22—41no—
329C22—41no—
330C22—41no—
331C22—41no—
332C22—41no—
333C22—41no—
334C22—41no—
335C22C2×#12011yescharacteristic
336C22#54611yescharacteristic
337C22#54711yescharacteristic
338C22C2×#13511yescharacteristic
339C22#57511yescharacteristic
340C22#57811yescharacteristic
341C22#57811yescharacteristic
342C4—22no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—42no—
347C4—42no—
348C4—42no—
349C4—42no—
350C4—42no—
351C4—81no—
352C4—81no—
353C4—81no—
354C4#52511yescharacteristic
355C4#53711yescharacteristic
356C4#54811yescharacteristic
357C4#55011yescharacteristic
358C2—42no—
359C2—81no—
360C2#240711yescharacteristic
361C2#300211yescharacteristic
362C2#301111yescharacteristic
363C2#330011yescharacteristic
364C2#332711yescharacteristic
365C2#333611yescharacteristic
366C2#335211yescharacteristic
subgroupG/H
center, socleC23C22×D4
commutator, Frattini, cosocleC22×C4C24
prime radicalC22×D4D4
LCSG ▷ C22×C4 ▷ C2 ▷ C1
UCSC1 ◁ C23 ◁ C2×C4×D4 ◁ G
LECSG ▷ C22×C4 ▷ C2 ▷ C1

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×#479C2semidirect
3. C2×C4×Q16C2semidirect
5. #2036C2semidirect
6. #2035C2semidirect
7. #2001C2semidirect
8. #2001C2semidirect
9. #1991C2semidirect
11. #1873C2semidirect
12. #1872C2semidirect
13. #1854C2semidirect
14. #1852C2semidirect
15. #1805C2semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
C2×#5522
C2×#4792
C2×C4×Q162
#25362
#20362
#20352
#20012
#19912
#19542
#18732
#18722
#18542
#18522
#18052